A 70.0 kg base runner begins his slide into second base when he is moving at a speed of 4.1 m/s. The coefficient of friction between his clothes and Earth is 0.70. He slides so that his speed is zero just as he reaches the base.
(b) How far does he slide?

We can solve this problem by using the work-energy principle. The kinetic energy lost by the runner as he slows down to rest is equal to the work done by the friction force.

### Given:
- Mass of the runner, \( m = 70.0 \, \text{kg} \)
- Initial speed, \( v_i = 4.1 \, \text{m/s} \)
- Final speed, \( v_f = 0 \, \text{m/s} \) (since he stops at the base)
- Coefficient of friction, \( \mu = 0.70 \)
- Gravitational acceleration, \( g = 9.8 \, \text{m/s}^2 \)

### Step 1: Calculate the initial kinetic energy

The initial kinetic energy of the runner is given by the formula:
\[
KE_i = \frac{1}{2} m v_i^2
\]
Substitute the values:
\[
KE_i = \frac{1}{2} \times 70.0 \times (4.1)^2
\]

### Step 2: Calculate the frictional force

The frictional force acting on the runner is:
\[
f_{\text{friction}} = \mu m g
\]
Substitute the values:
\[
f_{\text{friction}} = 0.70 \times 70.0 \times 9.8
\]

### Step 3: Work done by the friction force

The work done by the friction force is the product of the frictional force and the distance \( d \) over which it acts:
\[
W_{\text{friction}} = f_{\text{friction}} \times d
\]
Since the work done by friction is equal to the kinetic energy lost, we can equate the two:
\[
KE_i = f_{\text{friction}} \times d
\]
Solve for \( d \):
\[
d = \frac{KE_i}{f_{\text{friction}}}
\]

Now, let's compute this.The runner slides a distance of approximately 1.23 meters before coming to a stop.

Would you like further details or have any questions? Here are five related questions to expand on the concept:

1. How would the distance change if the coefficient of friction were smaller?
2. What would happen to the sliding distance if the runner’s mass increased?
3. How would the result differ if the runner were moving uphill or downhill?
4. How does the work-energy theorem apply in this scenario?
5. How can this problem be solved using Newton's second law?

**Tip:** Friction always opposes motion and converts kinetic energy into thermal energy or other forms, slowing down objects.

A 70.0 kg base runner begins his slide into second base when he is moving at a speed of 4.1 m/s. The coefficient of friction between his clothes and Earth is 0.70. He slides so that his speed is zero just as he reaches the base. (b) How far does he slide?

Calculate how far a 70 kg base runner slides into second base with an initial speed of 4.1 m/s and a coefficient of friction of 0.70. This solution applies the work-energy theorem and friction concepts, ideal for students in Grades 10-12.

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